| Exam Board | Edexcel |
|---|---|
| Module | F1 (Further Pure Mathematics 1) |
| Year | 2014 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Conic sections |
| Type | Parabola tangent intersection problems |
| Difficulty | Standard +0.8 This is a multi-part Further Maths parabola question requiring parametric tangent derivation, directrix knowledge, and coordinate geometry. While the tangent formula is standard for FM students, part (b) requires connecting the directrix equation with the given y-coordinate to find p, then using this to find D. Part (c) adds a straightforward area calculation. The question tests multiple concepts systematically but follows predictable FM patterns without requiring deep insight. |
| Spec | 1.07m Tangents and normals: gradient and equations |
8. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant.
The point $P \left( a p ^ { 2 } , 2 a p \right)$ lies on the parabola $C$.
\begin{enumerate}[label=(\alph*)]
\item Show that an equation of the tangent to $C$ at $P$ is
$$p y = x + a p ^ { 2 }$$
The tangent to $C$ at the point $P$ intersects the directrix of $C$ at the point $B$ and intersects the $x$-axis at the point $D$.
Given that the $y$-coordinate of $B$ is $\frac { 5 } { 6 } a$ and $p > 0$,
\item find, in terms of $a$, the $x$-coordinate of $D$.
Given that $O$ is the origin,
\item find, in terms of $a$, the area of the triangle $O P D$, giving your answer in its simplest form.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F1 2014 Q8 [12]}}